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Abstract

Exactly ergodicity in boundary-driven semi-infinite cellular automata (CA) are investigated. We establish all the ergodic rules in CA with 3, 4, and 5 states. We analytically prove the ergodicity for 18 rules in 3-state CA and 118320 rules in 5-state CA with any ergodic and periodic boundary condition, and numerically confirm all the other rules non-ergodic with some boundary condition. We classify ergodic rules into several patterns, which exhibit a variety of ergodic structure.

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1 Introduction

Consider a measure-preserving dynamical system ((\varOmega ,\mathcal {F},\mu ,\tau ^t)), where (\varOmega ) is a set, (\mathcal {F}) is a (\sigma )-algebra over (\varOmega ), (\mu ) is a probability measure, and (\tau ^t:\varOmega \rightarrow \varOmega ) is a one-parameter family of transformations that preserve (\mu ) and satisfy (\tau ^{t+s}=\tau ^t\circ \tau ^s) and (\tau ^0x=x) for any (x\in \varOmega ). Such a dynamical system is called ergodic if all invariant sets (A\in \mathcal {F}) that satisfy ((\tau ^t){-1}A=A) have the property (\mu (A)=0) or 1. One simple example is the linear motion on the N-dimensional torus (T^N=\mathbb {R}^N/\mathbb {Z}^N) written as (\tau ^t(\theta _1,\dots ,\theta _N)=(\theta +\omega _1t,\dots ,\theta _N+\omega _Nt)\mod 1), which is ergodic with respect to the uniform measure if and only if the frequencies (\omega ,\dots ,\omega _N) are rationally independent. This kind of motion appears as the flow on an invariant torus in a classical integrable Hamiltonian dynamical system. In this case, ergodicity is realized not in the whole space but in a sector restricted by a set of conserved quantities [1].

The notion of ergodicity was first introduced by Boltzmann [2] in order to justify the use of ensembles in statistical mechanics. This idea triggered a fruitful research field named ergodic theory [3, 4]. However, the existence of ergodic orbits is rarely proved except for several hard-ball systems [5]. For example, nearly integrable Hamiltonian systems are not ergodic due to the existence of invariant tori, which results from the Kolmogorov-Arnold-Moser (KAM) theorem [6]. As a KAM torus breaks up, chaotic region appears but in most cases it coexists with islands of stability, which mean invariant sets of positive measure composed of periodic chains of invariant tori. In the study of the standard map[7, 8], the abundance of the islands of stability is shown. On the other hand, there is no single parameter value for which ergodicity has been proved.

In reversible discrete-time dynamical systems with a finite phase space, every orbit must be periodic. Thus, ergodicity implies that the phase space is covered by only one periodic orbit. Rannou [9] studied reversible integer mappings obtained by coarse-graining a two-dimensional area-preserving mapping and found that it is not ergodic and the distribution of periodic orbits is similar to that of the random mappings.

In this paper, we discuss erodic orbits in cellular automata (CA). In conventional CA with finite volume, no single periodic orbit can cover the entire phase space because uniform states are mapped only to uniform states, that is, there is no path from uniform states to nonuniform states. However, if a CA has conserved quantities [10] and we consider a sector restricted by these conserved quantities, an ergodic orbit may reside on a subset specified by the conserved quantities. This is possible because some conservation laws in CA break the uniformity (translation invariance) of states. One example is rule 12R, a reversal variant of Wolfram’s elementary CA [11, 12]. Rule 12R, whose state at time t and site n is given by a pair of binary values ((\hat{x}_n^t,x_n^t)\in {(0,0),(0,1),(1,0),(1,1)}), is defined by the following equation

$$\begin{aligned} \hat{x}^{{t+1}_n=x_n^t,\quad x_n}{t+1}=(1-x_{n-1}^t)x_n^t-\hat{x}_n^t\mod 2. \end{aligned}$$

(1)

This rule has the following conservation law: If ((\hat{x}_n^0,x_n^0)=(0,0)) at the initial time, then they stay as ((\hat{x}_n^t,x_n^t)=(0,0)) at any time t. On the contrary, if ((\hat{x}_n^0,x_n^0)\ne (0,0)), then this site never takes ((\hat{x}_n^t,x_n^t)=(0,0)) for any t. Suppose that sites 0 and (N+1) have the value ((\hat{x},0,x_0)=(\hat{x}_{N+1}^0,x_{N+1}^0)=(0,0)) and sites 1 through N do not, which induces a corresponding sector with (3^N) states; ({(0,1),(1,0),(1,1)}^N). We numerically find and analytically prove that this sector contains only one periodic orbit. That is, rule 12R has ergodic orbits.

Observe that in rule 12R the dynamics of a cell at site n depends only on itself and its left cell ((n-1)) and does not depend on its right cell ((n+1)). Namely, this CA is driven from left to right, and this driving induces ergodic dynamics persisting any length N. This ergodic behavior of rule 12R motivates us to investigate a semi-infinite reversible cellular automaton where site 1 periodically evolves and the time-evolution of site (n>1) is given by a permutation determined by the state of site (n-1). In rule 12R, site 1 evolves with period 3 like ((\hat{x}_1,x_1)=(1,1)\rightarrow (1,0)\rightarrow (0,1)\rightarrow (1,1)), because (x_0^t=0) for any t. The time evolution of site n is given by

$$\begin{aligned} (\hat{x}_n^{t+1},x_n^{t+1})= {\left{ \begin{array}{ll} (1,1) & {\textrm{if}};(\hat{x}_n^t,x_n^t)=(0,1) \ (0,1) & {\textrm{if}};(\hat{x}_n^t,x_n^t)=(1,0) \ (1,0) & {\textrm{if}};(\hat{x}_n^t,x_n^t)=(1,1) \end{array}\right. } \end{aligned}$$

(2)

if (x_{n-1}^t=0), and

$$\begin{aligned} (\hat{x}_n^{t+1},x_n^{t+1})=(x_n^t,\hat{x}_n^t) \end{aligned}$$

(3)

if (x_{n-1}^t=1). They are permutations. Moreover, because the rule does not depend on the right nearest neighbor, site (N+1) plays the role only restricting the system size.

A rule of CA where the dynamics of a cell depends only on itself and its left cells is known as a one-way or directed CA. One-way CAs are studied with different boundary conditions from ours [13]. One-way stochastic CAs are also well-studied. An example is the East model[14,15,16,17], which is a kinetically constrained Ising model to model slow dynamics in glassy systems. Many-body localization and quantum ergodicity have been discussed with its quantum variants[18, 19].

In this paper, we restrict ourselves to classical and deterministic semi-infinite one-way CAs and clarify the condition for the existence of ergodic orbits. We thoroughly examine semi-infinite CA with 3, 4, and 5 states. We numerically find and analytically prove that there are 18, 0, and 118320 ergodic rules in CA with 3, 4, and 5 states, respectively. The 18 ergodic rules in CA with 3 states are classified into 2 types, both of which are proven to be ergodic. The ergodicity of rule 12R is also proved in this context. The 118320 rules in CA with 5 states are classified into 72 types with 206 subtypes, all of which can be proven to be ergodic. Other rules are numerically shown to be non-ergodic. In other words, we succeed in determining all ergodic rules in one-way CA with 3, 4, and 5 states. As far as we know, our study is the first exhaustive classification of semi-infinite one-way classical deterministic CAs with periodic driving of the leftmost cell.

This paper is organized as follows. In Section. 2, we introduce definitions and notation in terms of symmetric groups. In Section. 3, we prove all ergodic rules in 3-state CA in detail, which plays as a good introduction to the proof of ergodicity in 5-state CA. In Section. 4, we treat 4-state CA, which has no ergodic rules. In Section. 5, we summarize the results on 5-state CA, and present the proof strategy and symbols employed in the proof. In Section. 6, we prove the ergodicity of rules in 5-state CA. We classify rules into 5 patterns with some sub-categories, and present proofs for several representative rules. In Section. 7, we consider generalizations of petterns to CA with more than 5 states. In Section. 8, we list several open problems.

We list all the subtypes of ergodic rules and their transition maps in 5-state in Sec. 5.1. The structure of these ergodic rules, which play a crucial role in proving their ergodicity, is presented in Appendix. A.

2 Semi-infinite reversible cellular automata

2.1 Symmetric group

For completeness, we first describe some basic facts of symmetric groups. The symmetric group (S_k) is composed of all bijections from ({0,1,\dots ,k-1}) to itself. The elements of (S_k) are called the permutations. Each permutation (\pi \in S_k) is specified in the two-line form (\displaystyle \begin{pmatrix}0 & 1 & \cdots & k-1\ \pi (0) & \pi (1) & \cdots & \pi (k-1)\end{pmatrix}). A permutation (\pi ) is called a p-cycle or a cycle of length p if (\pi ^p(x)=x) and (\pi ^q(x)\ne x) for (0\le q<p). In the cycle notation, a p-cycle (\pi ) is expressed by the enumeration of states as ((x\ \pi (x)\ \pi ^2(x) \dots \pi ^{p-1}(x))). Every permutation is represented by a product of non-overlapping cycles. An example of the cycle notation is

$$\begin{aligned} \begin{pmatrix} 0 & 1 & 2 & 3 & 4 & 5 \ 1 & 3 & 4 & 0 & 2 & 5 \end{pmatrix}=(0,1,3)(2,4). \end{aligned}$$

(4)

In this paper, we omit 1-cycles from the cycle notation for brevity. Let (n_k) be the numbers of k-cycles in (\pi ), and using this we introduce the cycle type of (\pi ) as (1^{n_1} \ 2^{n_2}\cdots k^{n_k}), where we can omit the factors with (n_k=0). In this expression, the superscript of m represents the number of m-cycles in this permutation. For example, the cycle type of the permutation in Eq. (4) is (2^13^1). We call permutations (\pi ) and (\sigma ) conjugate if they satisfy (\pi =\tau \sigma \tau ^{-1}) for some (\tau ). Two permutations are conjugate if and only if they have the same cycle type. The permutations with the same cycle type compose a conjugacy class.

We encode each permutation (\pi \in S_k) by a natural number as

$$\begin{aligned} n(\pi )=\sum _{i=0}^{k-2}r(\pi ,i)(k-1-i)!, \end{aligned}$$

(5)

where (r(\pi ,i)=\left| {i\le j\mid \pi (j)\le \pi (i)}\right| -1), and (\left| A\right| ) means the number of elements in set A. Noting (0\le r(\pi ,i)\le k-i-1), we confirm that this encoding provides a one-to-one correspondence between k! possible permutations and integers (0\le n\le k!-1). In this paper, we specify permutations by using integers with this encoding.

In the case of (k=3), we have the following 6 permutations

$$\begin{aligned} \sigma _0&=\begin{pmatrix}0 & 1 & 2\ 0 & 1 & 2\end{pmatrix}=\textrm{id}. ,&\sigma _1&=\begin{pmatrix}0 & 1 & 2\ 0 & 2 & 1\end{pmatrix}=(1,2),&\sigma _2&=\begin{pmatrix}0 & 1 & 2\ 1 & 0 & 2\end{pmatrix}=(0,1), \end{aligned}$$

(6)

$$\begin{aligned} \sigma _3&=\begin{pmatrix}0 & 1 & 2\ 1 & 2 & 0\end{pmatrix}=(0,1,2),&\sigma _4&=\begin{pmatrix}0 & 1 & 2\ 2 & 0 & 1\end{pmatrix}=(0,2,1),&\sigma _5&=\begin{pmatrix}0 & 1 & 2\ 2 & 1 & 0\end{pmatrix}=(0,2). \end{aligned}$$

(7)

As seen from this, (S_3) is decomposed into the sum of conjugacy classes; ((1^3)={\sigma _0}), ((1^12^1)={\sigma _1,\sigma _2,\sigma _5}), and ((3^1)={\sigma _3,\sigma _4}).

In the case of (k=5), we have 120 permutations listed in Table. 1:

Table 1 Permutations (\sigma _i\in S_5). Each row shows i, (\sigma _i(0)\sigma _i(1)\sigma _i(2)\sigma _i(3)\sigma _i(4)) and its cycle notation

2.2 Semi-infinite reversible CA

Throughout this paper, we consider the semi-infinite reversible cellular automata with k states defined by the following time-evolution rules:

$$\begin{aligned} x_1^{t+1}&=\pi _\textrm{B}(x_1^t) \end{aligned}$$

(8)

$$\begin{aligned} x_n^{t+1}&=\pi _{x_{n-1}^t}(x_n^t),;;(n\ge 2) \end{aligned}$$

(9)

where (x_n^t) takes values in the set ({0,1,\dots ,k-1}), (\pi _\textrm{B}) is a fixed cycle of length k, and (\pi _x), ((x\in {0,1,\dots ,k-1})) is a permutation. The fixed permutation (\pi _\textrm{B}) in the first line is the boundary driving on (n=1). The second line means that the permutation acting on site (n(\ge 2)) at time t is determined by the state on site (n-1) at the same time t.

The reversibility of the dynamics is evident because (x_1^t=\pi _\textrm{B}^{-1}(x_1{t+1})), and (x_n^t) is inductively obtained from ((x_1^{t+1},\dots ,x_n^{t+1})) by using the relation (x_n^{t}=\pi _{x_{n-1}^t}{-1}(x_n^{t+1})). The rule of CA is specified by (\pi _\textrm{B}) and a set of k permutations ((\pi _0,\dots ,\pi _{k-1})). If (\pi _{s}) ((0\le s\le k-1)) is encoded by (i_s), we also refer to the rule as ((i_0,\dots ,i_{k-1})) (with boundary dynamics (\pi _\textrm{B})). Thus, there are ((k!)^k) rules and ((k-1)!) boundary dynamics in the k-state semi-infinite reversible cellular automata.

The phase space of site 1 through n consists of (k^n) states. Since the sites (m>n) do not affect the dynamics of those n sites, the sites (1\le i\le n) can be considered as a closed dynamical system. We say that site n is ergodic if it evolves with period (k^n) and has no shorter period, and that the rule is ergodic if every site is ergodic. Since (\pi _\textrm{B}) is a k cycle, site 1 is ergodic for any rule by construction. However, the ergodicity of sites (n\ge 2) depends on the rule.

Given a temporal sequence on site n between (t_0\le t\le t_1) as (x_n^{t_0}x_n^{t_0+1}\cdots x_n^{t_1-1}), the transformation on site (n+1) from (t=t_0) to (t=t_1) denoted by (T_{n+1}^{{t_0\rightarrow t_1}: x_{n+1}}{t_0}\mapsto x_{n+1}^{t_1}) is determined by the product of permutations as

$$\begin{aligned} T_{n+1}^{{t_0\rightarrow t_1}=\pi _{x_n}{t_1-1}}\pi _{x_n^{t_1-2}}\cdots \pi _{x_n^{t_0+1}}\pi _{x_n^{t_0}} \end{aligned}$$

(10)

Thus, (T_{n+1}^{{t_0\rightarrow t_1}) itself is a permutation. Under the assumption of the ergodicity of site n, the two statements that site (n+1) is ergodic and that permutation (T_{n+1}}{0\rightarrow k^n}) is a k-cycle are equivalent with each other. It is because sequence (x_{n+1}^{0\rightarrow x_{n+1}}{k^n}\rightarrow x_{n+1}^{{2k^n}\rightarrow \cdots \rightarrow x_{n+1}}{k^{n+1}}) is generated by repeated application of (T_{n+1}^{0\rightarrow k^n}).

3 CA with 3 states

For (k=3), there are (6^3=216) rules and two boundary dynamics. We first numerically examine the ergodicity of these rules with boundary dynamics (012).

As a result of numerical simulation, we find that the following two types of rules can be ergodic. The rule in type 1 is illustrated by the state transition diagram depicted in Fig. 1. In the state transition diagram, the arrow from x to y with index z represents the rule (\pi _z(x)=y). An asterisk stands for any values (i.e., (*={ a,b’,c’}) in this case). We set the symbols ((a,b,c)) as (0, 1, 2) or its permutation, and set ((b’,c’)=(b,c)) or ((c,b)). In the case of ((a,b,c)=(1,0,2)) and ((b’,c’)=(c,b)=(2,0)), as an example, the permutations are (\pi _1=\pi _2=(02)) and (\pi _0=(102)). We have 12 rules in this type. If we identify (a=(1,1)), (b=(1,0)), (c=(0,1)), and ((b’,c’)=(c,b)), the rule corresponds to rule 12R on the sector ({ (0,1), (1,0), (1,1)}^{\mathbb {N}}). The rule in type 2 is illustrated in Fig. 2. Because the exchange of (b) and (c) does not change the diagram, there are 6 rules in this type.

Fig. 1

Type 1 ergodic rules for (k=3). The state transition diagram and the rules that belong to this type. Rule ((\pi _0,\pi _1,\pi _2)=(\sigma _i,\sigma _j,\sigma _k)) is represented as (i, j, k) in the Table

Fig. 2

Type 2 ergodic rules for (k=3)

Both types’ rules are numerically confirmed to keep ergodic using boundary dynamics (\pi _\textrm{B}=(021)). In the following, we shall prove the ergodicity of the two types of rules for both boundary dynamics (\pi _\textrm{B}=(abc)) and (\pi _\textrm{B}=(acb)).

(\underline{\text {Proof}_{\text {of}}\text {ergodicity}~\text {of}\text {Type 1}}): Type 1 ergodic CA consists of the permutations (\pi _a=\pi _b=(bc)) and (\pi _c=(abc)) for ((b’,c’)=(b,c)), or (\pi _a=\pi _c=(bc)) and (\pi _b=(abc)) for ((b’,c’)=(c,b)). For (n\ge 2), we can prove by induction the following two propertiesFootnote 1 (see an example of sequences in the first, second, and third cells shown in Fig. 4):

1. 1.

Site n has period (3^n).

1. 2.

The sequence of site n, ({x_n^t\mid 0\le t\le 3^n-1}) is a concatenation of one a and units of bc and aa.

Unit bc at site (n\ge 2) can be understood from the facts that b is necessarily mapped to c and that the preimage of state c is b only. Thus, only the behavior of a’s is nontrivial.

We now start mathematical induction. We can directly confirm that site 2 satisfies these two properties for both boundary rules. Now we assume that site n satisfies these two properties. Then, property 1 suggests that there must be the same number of as, bs, and cs in the sequence ({x_n^t\mid 0\le t\le 3^n-1}). Accordingly, the sequence in site n consists of a single a, ((3^{n-1}-1)/2) aa’s, and (3^{n-1}) bc’s. Then, in the transformation from (x_{n+1}^{0) to (x_{n+1}}{3^n}) (i.e., (T_{n+1}^{{0\rightarrow 3^n})), we apply (\pi _a) once, (\pi _a^2) by ((3}{n-1}-1)/2) times, and (\pi _c\pi _b) by (3^{n-1}) times. Using the relations (\pi _a^2=\textrm{id}) and (\pi _c\pi _b) is a transposition, we find

$$\begin{aligned} T_{n+1}^{0\rightarrow 3^n}={\left{ \begin{array}{ll} \pi _c\pi _b\pi _a& \textrm{or} \ \pi _a\pi _c\pi _b.& \end{array}\right. } \end{aligned}$$

(11)

In either case, it is a cycle of length 3. This means that site (n+1) has period (3^{n+1}), which is property 1.

Next, we prove that site (n+1) satisfies property 2. To this end, in the following, we treat the case of ((b’,c’)=(b,c)) for convenience of explanation. A similar argument holds for ((b’,c’)=(c,b)). Consider a temporal region of site n consisting of units bc and aa. Then, the corresponding temporal region of site (n+1) also consists of units bc and aa, because (\pi _a\pi _a) induces transitions

$$\begin{aligned} a\xrightarrow {a}a&\xrightarrow {a} a, \ b\xrightarrow {a}c&\xrightarrow {a} b, \ c\xrightarrow {a}b&\xrightarrow {a} c, \end{aligned}$$

and (\pi _{c}\pi _{b}) induces transitions

$$\begin{aligned} a\xrightarrow {b}a&\xrightarrow {c} b, \ b\xrightarrow {b}c&\xrightarrow {c} a, \ c\xrightarrow {b}b&\xrightarrow {c} c. \end{aligned}$$

As clearly seen from these transitions, if the initial state of site (n+1) is c, then both (\pi _c\pi _b) and (\pi _a\pi _a) induce repeated bc as (c\rightarrow b\rightarrow c\rightarrow b\rightarrow \cdots ). On the other hand, if the initial state of site (n+1) is a or b, then (\pi _a\pi _a) keeps the unit aa and bc in site (n+1), while (\pi _c\pi _b) switches these two units, (aa\leftrightarrow bc), in site (n+1) (See also Fig. 3). In addition, single (\pi _a) is transposition (bc), which plays the role of switching the initial state between b and c with keeping the structure of units. Note that single (\pi _a) acts on state a exactly once in a single period with length (3^{n+1}), and when single (\pi _a) applies to state a, this plays the role of adding a single a in site (n+1), which results in a single a besides unit aa in the sequence in site (n+1) with length (3^{n+1}). The overall argument confirms property 2.

Fig. 3

The flow chart of the dynamics of Type 1 CA with ((b’,c’)=(b,c)). If the state is a or b, the application of (\pi _c\pi _b) and (\pi _a\pi _a) provides units bc and aa. In contrast, if the state is c, the application of (\pi _c\pi _b) and (\pi _a\pi _a) always provides (c\rightarrow b\rightarrow c\rightarrow b\rightarrow c\rightarrow \cdots ). A single (\pi _a) switches these two phases if the state is b or c

Fig. 4

The dynamics of a Type-1 rule with ((a,b,c)=(0,1,2)) and ((b’,c’)=(b,c)=(1,2)). The first, second, and third lines have period 3, (9=3^2), and (27=3^3)

(\underline{\text {Proof}_{\text {of}}\text {ergodicity}_{\text {of}}\text {Type 2}}): Type 2 ergodic CA consists of (\pi _a=\textrm{id}), (\pi _{b’}=(ab)), and (\pi _{c’}=(ac)). We can fix the initial value (x_n^0=a) without loss of generality. If we prove ergodicity for site (n+1) in this case, it means sites 1 through (n+1) evolves periodically with period (3^{n+1}), where every pattern of values appear only once. Thus, any other choice of initial condition is included in the period.

We prove the following two properties by mathematical induction (see an example of sequences in the first, second, and third cells shown in Fig. 5):

1. 1.

Site n has period (3^n).

1. 2.

The sequence ({x_n^{t}\mid 0\le t\le 3^n-1}) is divided into two intervals. One consists of a and b only, and the other consists of a and c only.

Site 1 clearly satisfies both properties 1 and 2. Now we assume that site n satisfies both properties 1 and 2. Combining property 2 and the fact that (\pi _a) is the identity, we find

$$\begin{aligned} T_n^{3^n}={\left{ \begin{array}{ll} \pi _{b’}^{3{n-1}}\pi _{c’}^{3{n-1}}=\pi _{b’}\pi _{c’}& \textrm{or} \ \pi _{c’}^{3{n-1}}\pi _{b’}^{3{n-1}}=\pi _{c’}\pi _{b’}.& \end{array}\right. } \end{aligned}$$

(12)

In either case, it is a 3-cycle, and accordingly, site (n+1) has period (3^{n+1}), which implies property 1.

To prove property 2, we look into the dynamics more in detail. In the interval of site n with a or (b’) only, site (n+1) either takes “a or b only" or stays at c, because (\pi _a) is the identity and (\pi _{b’}) is a transposition between a and b. For a similar reason, in the interval of site n with a or (c’) only, site (n+1) either takes “a or c only" or stays at b. These facts assure that property 2 holds for site (n+1), which is exhibited in Fig. 6.

Fig. 5

The dynamics of a Type-2 rule with ((a,b,c)=(0,1,2)) and ((b’,c’)=(b,c)=(1,2)). The first, second, and third lines have period 3, (9=3^2), and (27=3^3)

Fig. 6

We draw transitions in three periods in the case of (T_n^{3^n}=\pi _{c’}\pi _{b’}). It directly shows that if property 2 is satisfied in site n, it is also satisfied in site (n+1)

4 CA with 4 states

We here examine all the boundary-driven semi-infinite reversible CA with 4 states. It is numerically confirmed that there are no ergodic rules in the case (k=4). The period of site 2 is shorter than (4^2) in any rule and any boundary dynamics.

5 CA with 5 states: summary of results

5.1 List of ergodic CA

We here examine all the boundary-driven semi-infinite reversible CA with 5 states. In the case (k=5), there are 120 permutations listed in Table 1. The semi-infinite reversible CA of (k=5) has ((5!)^5=24883200000) rules and (4!=24) boundary rules. In order to moderate the number of ergodic rules, we restrict ourselves to the rules which are ergodic for any boundary rules. In the following, we use the term ergodic in this sense.

We numerically searched candidates of ergodic rules with increasing system size N. Namely, if a rule generates an orbit of period (5^N) for every boundary dynamics, it is a candidate. The numbers of the candidates for N are exhibited in Table 2. Though it looks convergent at (N=11), true convergence is reached at (N=15), where we have 118320 rules. Thanks to the boundary condition, if a rule is ergodic, its conjugate rules are all ergodic. Thus, we can classify the ergodic rules with the numbers of rules in conjugacy classes into 72 types as Table 3. Further, we classify the rules in subtypes by their state transition diagrams. The list of ergodic rules classified into subtypes is presented in the form of state transition diagrams in Fig. 7 with the parameter choice shown in Table 4. Notice that although only representative rules are shown, their conjugate rules are also ergodic. The number of ergodic rules includes those conjugate rules.

The remainder of this paper is devoted to the proof of ergodicity of these rules.

Table 2 The numbers of candidates for ergodic rules for system size N. It is analytically proven that the number of candidates 118320 at (N=15) is true convergence

Table 3 Types of the (k=5) ergodic rules classified by the conjugacy classes and the number of ergodic rules in each type

Table 4 Ergodic rules for (k=5). ({i_1,i_2}) means that any order of the elements is accepted. The semicolon represents a separation of rules

5.2 Proof strategy

Our proof of ergodicity takes the form of mathematical induction. Given the sequence in site n with period (5^n). Let (\varSigma =T_{n+1}^{0\rightarrow 5^n}) be the permutation in site (n+1) through (5^n) steps (one period in site n), which is induced by the sequence in site n. In our mathematical induction, we show that

1. (a)

Site n has period (5^n) (i.e., (\varSigma ) is a 5-cycle).

1. (b)

The sequence of site n satisfies some structure.

The former fact implies that site n is ergodic. The form of the structure is specified on each rule of the CA, which will be explained later.

Fig. 7

State transition diagrams of the ergodic rules for (k=5). The arrow’s index shows the values of left nearest neighbor. An asterisk means any value. Superscript c means complement. For example, ((12)^c) means 034

The above two properties are shown to be satisfied in site (n=2) by a direct computation, and therefore the remainder of this paper is devoted to proving the aforementioned two properties in site (n+1) under the assumption that they are satisfied in site n. Since it requires extremely many pages to fill the proofs for all possible rules, and most of the proofs employ similar techniques, we classify the structures of all rules into several patterns and provide proofs only for some representative rules. We list structures of all rules explicitly in Appendix. A, with which a reader can prove the ergodicity of each rule without much effort if one needs it.

5.3 Symbols

To present the structures of CA, We use the following terminology:

(\underline{\text {period}~\text {sequence}}): a sequence of (5^n) symbols (i.e., one period) in site n.

(\underline{\text {unit}})A sequence of a few symbols (usually two or three symbols) is called unit. See the explanation on Pattern B (Sec. 6.3) for details and examples.

(\underline{\text {island}}): We decompose a period sequence into several large regions. A single long region is called island. See the explanation on Pattern A (Sec. 6.2) for details and examples.

To describe structures with units, we introduce the following symbols:

([\cdot ])representing a unit. The symbols in the bracket appear in its order.

(\lfloor \cdot \rfloor _\textrm{even})representing a block with even length. For example, (\lfloor 0,3\rfloor _\textrm{even}) might be 00, 0303, 333033.

(\lfloor \cdot \rfloor _\textrm{odd})representing a block with odd length. For example, (\lfloor 0,3\rfloor _\textrm{odd}) might be 033, 0, 303, 00300.

(\lfloor \cdot \rfloor _\textrm{odd}^1)meaning that this block appears only once in one period sequence with (5^n) symbols. The superscript 1 in (\lfloor \cdot \rfloor _\textrm{even}^1), (\langle \cdot \rangle _\textrm{even}^1), and (\langle \cdot \rangle _\textrm{odd}^1) represents the same property.

In addition, to describe structures with islands, we introduce the following symbol:

((\cdot ))representing an island. The symbols in the bracket can appear in the island. For example, (0, 1, 2) means that only 0, 1, and 2 may appear in this island (i.e., 3 and 4 never appear). The island (0, [12]) can be 001200 and 1212120, but neither 001020 nor 0112200.

To grasp the ideas of islands and units, we here present several examples of structures.

The structure of Type09-2 (shown in Sec. 6.2.1) employs islands as (0, 1, 2, 3)(0, 4), stating that the symbol sequence of length (5^n) is decomposed into two regions; that consisting of 0, 1, 2, and 3, and that consisting of 0 and 4. The first, second, and third sequences are expressed as (see also Fig. 9 in page 29)

1. 1.

01234 (\rightarrow ) 0123 and 4

1. 2.

0000444440111212221233333 (\rightarrow ) 0000444440 and 111212221233333

1. 3.

0000012301111122121121212111112301222221121221212122222301233333333333

3333333333012300000000000040404444444444444444444440404

(\rightarrow ) 000001230111112212112121211111230122222112122121212222230123333333

3333333333333301230 and 0000000000040404444444444444444444440404

As shown above, the symbol sequences are decomposed into (0, 1, 2, 3) (i.e., a region with 0, 1, 2, and 3) and (0, 4) (i.e., a region with 0 and 4). This is a typical structure with islands. We note that the beginning and the end of islands have some ambiguity. In the second sequence in the above case, the decomposition into two islands can also be “44444 and 01112122212333330000" and “0044444 and 011121222123333300". Although a consistent choice of islands is helpful for visibility, the ergodicity holds regardless of the choice of islands.

The structure of Type48-2 (shown in Sec. 6.3.1) employs units as ([12], [00], \lfloor 3,4\rfloor _\textrm{even}, \lfloor 0\rfloor _\textrm{odd}^1), stating that the symbol sequence of length (5^n) consists of (many) 12, 00, an even length region only with 3 and 4, and a single sequence of 0 with odd length. The first, second, and third sequences are expressed as (see also Fig. 20 in page 37)

1. 1.

01234 (\rightarrow ) 0/12/34

1. 2.

0012121212123444343334000 (\rightarrow ) 00/12/12/12/12/12/3444343334/000

1. 3.

0001234001234444444444434343434343434333333333334343400123400121212121 2121212121212121212121212121212123400123400000000000000

(\rightarrow 000/12/ 34/00/12/344444444444343434343434343333333333343434/00/12/34/)

00/12/12/12/12/12/12/12/12/12/12/12/12/12/12/12/12/12/12/12/12/12/34/00/

12/34/00/00/00/00/00/00/00

As shown above, the symbol sequences consist of 12, 00, an even length region only with 3 and 4, and a single sequence of 0 with odd length. This is a typical structure with units.

In addition to above, to describe more complicated and exotic structures, we further introduce some terms as

(\underline{\text {block}~\text {unit}}): A finite length of symbol sequence is called block unit, if what symbols or units can appear in this sequence is determined while its order is not determined (see the explanation on (\lfloor \cdot \rfloor ) below). In several cases, we need not to distinguish units and block units. In these cases, we call block units also as simply unit.

(\underline{\text {queue}}): A finite length of symbol sequence is called queue, if the order of symbols or units in this sequence is determined (see the explanation on (\langle \cdot \rangle ) below).

and symbols as

({ \cdot }): representing a symbol which is one of the symbols in the bracket. For example, ({ 0,2,3}) is a single symbol of 0, 2, or 3. ({0, [12]}) is one of symbol 0 or unit 12.

(\langle \cdot \rangle _\textrm{even})representing a queue with even number of symbols. The “(+)" sign concatenates symbols. For example, (\langle 2+{3,4}\rangle _\textrm{even}) might be 2324242423, 4232323232, where 2 and “3 or 4" appears alternately. (\langle \cdot \rangle _\textrm{odd}) represents a queue with an odd number of symbols.

(*): The superscript (*) (e.g., (\lfloor 0,1\rfloor _\textrm{even}^*)), which is used only in Pattern D-2-2, means that the rule (even or odd) might be violated at the edge of islands. See the section for Pattern D-2-2 for details.

Furthermore, to explain the dynamics, we distinguish transition and permutation itself by taking an example. Let (\pi _1=(12)(34)) and (\pi _4=(23)). Then, the permutation (\pi _{4141}=\pi _4\pi _1\pi _4\pi _1) simply means (14)(23). If the initial state is 1, then the permutation (\pi _{4141}) conveys state 1 to state 4. On the other hand, if the initial state is 1, the transition induced by (\pi _{4141}) is (1\rightarrow 2\rightarrow 3 \rightarrow 4), where we take the path into account. Thus, the permutation (\pi _{4141}) is inside ({1,4}) and ({2,3}), while the transition induced by (\pi _{4141}) is not inside ({1,4}) and ({2,3}) (because a state inside ({1,4}) goes out from this set in the middle of the transition).

6 CA with 5 states: Patterns and proofs

6.1 Classification

We here briefly summarize the classification of the rules of ergodic CA with 5 states. Details are described in the corresponding subsections.

We classify the rules of ergodic CA into five patterns from Pattern A to Pattern E.

Pattern A employs several islands in its structure, which can be understood as a generalization of type-2 in 3-state CA. This pattern is further classified into three subclasses, A-1, A-1-2, and A-2. Although A-1 and A-1-2 are simple, A-2 is complex such that the proof of ergodicity of A-2 requires additional elaborated ideas.

Pattern B employs several units in its structure, which can be understood as a generalization of type-1 in 3-state CA. This pattern is further classified into three subclasses, B-1, B-2, and B-3. Although B-1 is simple, the structure of B-2 and B-3 are highly complicated.

Pattern C newly appears in 5-state CA, where states are decomposed into two sets, and states in these two sets appear alternatingly.

Pattern D is a hybrid of Pattern A and Pattern B. This pattern is further classified into five subclasses, D-1, D-2, D-2-2, D-2-3, and D-3. The first three, D-1, D-2, D-2-2, are relatively simple, though D-2-3 and D-3 have highly complicated structures.

Pattern E is highly exceptional, and only few rules belong to it.

6.2 Pattern A: divided by several islands

6.2.1 A-1: The simplest case

The structure in pattern A-1 consists of several islands (I_1, I_2,\ldots ) on 5 states. The rule in pattern A-1 satisfies the following properties:

1. (a).

States a, b belonging to the same island (i.e., (a,b\in I_i) for some i) provide commutative permutations; (\pi _a\pi _b=\pi _b\pi _a).

1. (b).

State x can be mapped onto state y by some permutation (\pi _a) only if x and y belong to the same island.

Thanks to the commuting condition (a), the computation of the permutation with one period (\varSigma ) becomes simple.

We shall demonstrate how to prove ergodicity and the structure by taking several representatives. The structures of the remaining rules are presented in Appendix. A.

(\underline{\hbox {Type03};_(a,b)=(2,4):(\pi _0=\pi _1=\pi _3=\textrm{id},_{\pi _2=(012),}\pi _4=(034))})

The structure of this rule is (0, 1, 2)(0, 3, 4), which consists of two islands, (I_1=(0,1,2)) and (I_2=(0,3,4)). We can confirm condition (b) by just drawing its transition map (see Figure. 7 in Sec. 5.1). Since (\pi _0, \pi _1, \pi _3) are identity maps, we see that the commuting condition (a) is satisfied.

Let (I_1^{[n]}) and (I_2^{[n]}) be the islands in the period sequence in site n. Remark that the bracket ([\cdot ]) in the superscript represents the position of site, and does not represent a unit. By the assumption from induction, the period sequence with length (5^n) in site n consists of (I_1^{[n]}) and (I_2^{[n]}), which are sequences of symbols only in (I_1) and (I_2), respectively. We denote by (\pi _{I_1^{[n]}}) and (\pi _{I_2^{[n]}}) the transitions on site (n+1) induced by islands (I_1) and (I_2) in site n. The total transition in site (n+1) in a single period (5^n), (\varSigma =T^{0\rightarrow 5^n}_{n+1}), can be written as ((\pi _4)^{5{n-1}} (\pi _2)^{5{n-1}}=(034)^{5{n-1}}(012)^{5{n-1}}), where ((034)) and ((012)) represent 3-cycles (\begin{pmatrix}0& 3& 4\ 3& 4& 0 \end{pmatrix}) and (\begin{pmatrix}0& 1& 2 \ 1& 2& 0 \end{pmatrix}), respectively.

Since we have

$$\begin{aligned} \pi _{I_2^{[n]}}&=(\pi _4)^{5{n-1}}=(034)^{5{n-1}}= {\left{ \begin{array}{ll} (043) & n \ \mathrm{is \ even}, \ (034) & n \ \mathrm{is \ odd}, \end{array}\right. } \end{aligned}$$

(13)

$$\begin{aligned} \pi _{I_1^{[n]}}&=(\pi _2)^{5{n-1}}=(012)^{5{n-1}}= {\left{ \begin{array}{ll} (021) & n \ \mathrm{is \ even}, \ (012) & n \ \mathrm{is \ odd}, \end{array}\right. } \end{aligned}$$

(14)

the permutation on site (n+1) for one period of site n is a 5-cycle:

$$\begin{aligned} \varSigma =\pi _{I_2^{[n]}}\pi _{I_1^{[n]}}= {\left{ \begin{array}{ll} (043)(021)=(02143) & n \ {\textrm{is even}}, \ (034)(012)=(01234) & n \ {\textrm{is odd}}, \end{array}\right. } \end{aligned}$$

(15)

which guarantees the ergodicity in site (n+1). Furthermore, the structure (0, 1, 2)(0, 3, 4) is inherited from site n to site (n+1), because the state in island (I_1) is transferred to island (I_2) only when the state after (\pi _{I_1^{[n]}}) is 0 (see Fig. 8).

Fig. 8

The sequence of first, second, and third cells in Type03 with ((a,b)=(2,4)). The sequence in the third cell is divided into 5 lines that go from the top to the bottom. The light gray and dark gray regions represent islands (I_1) and (I_2), respectively. Two solid rectangles represent the initial (resp. final) state and the final (resp. initial) state of permutation (\pi _{I_1^{[n]}}) (resp. (\pi _{I_2^{[n]}})) for (n=2)

Since this is our first example, we shall examine the last point (inheritance of structure) in detail. Owing to ergodicity, (\pi _{I_1^{[n]}}) restricted to (I_1) serves as a cyclic permutation in (I_1), and (\pi _{I_2^{[n]}})restricted to (I_2) serves as a cyclic permutation in (I_2). In addition, (\pi _{I_1^{[n]}}) acts trivially on states 3 and 4, and (\pi _{I_2^{[n]}}) acts trivially on states 1 and 2. Now we can demonstrate the inheritance of structure as follows: Suppose that the initial state is 0. First, (\pi _{I_1^{[n]}}) applies, which induces a transition inside island (I_1), and the resulting state is not 0 (i.e., 1 or 2) due to ergodicity. Next, (\pi _{I_2^{[n]}}) applies, while it keeps state 1 as it is, and thus the state remains inside island (I_1). Then, (\pi _{I_1^{[n]}}) applies, which induces a transition inside island (I_1), and the resulting state is still 1 or 2 due to ergodicity. Further, (\pi _{I_2^{[n]}}) applies again, while it keeps state 1 as it is, and thus the state remains inside island (I_1). Finally, (\pi _{I_1^{[n]}}) applies, which induces a transition inside island (I_1), and the resulting state is now 0. Thus, the next application of (\pi _{I_2^{[n]}}) induces a nontrivial transition inside (I_2), and the resulting state is not 0 (i.e., 3 or 4) due to ergodicity. The remaining argument is similar to the above, with exchanging the role of (I_1) and (I_2). In summary, the state remains inside (I_1) for two and a half periods, and remains inside (I_2) for another two and half periods, which implies the inheritance of the structure.

In most of type-A rules, the inheritance of the structure (i.e., the structure in site n guarantees the same structure in site (n+1)) can be shown in the aforementioned manner with the help of ergodicity. Therefore, in the following, we mainly investigate the ergodicity, and briefly comment on the inheritance of the structure.

(\underline{\hbox {Type09-2};_(a,b)=(2,3):(\pi _0=\pi _1=\textrm{id},_{\pi _2=(12),} \pi _3=(04)(12),~\pi _4=(0123))})

The structure of this rule is (0, 1, 2, 3)(0, 4), which consists of two islands, (I_1=(0,1,2,3)) and (I_2=(0,4)). We can confirm condition (b) by just drawing its transition map (see Figure. 7 in Sec. 5.1). Observe that (\pi _0, \pi _1) are identity maps and (\pi _2=\pi _3) behave as (12) in (I_1), and that (\pi _0, \pi _1, \pi _2, \pi _4) are identity maps in (I_2), which ensure the commuting condition (a).

The maps (\pi _{I_1^{[n]}}) and (\pi _{I_2^{[n]}}) are calculated as

$$\begin{aligned} \pi _{I_1^{[n]}}&=(\pi _2)^{5{n-1}}(\pi _3)^{5{n-1}}=(12)^{{2\cdot 5}{n-1}}(04)^{5{n-1}}=(04), \end{aligned}$$

(16)

$$\begin{aligned} \pi _{I_2^{[n]}}&=(\pi _4)^{5{n-1}}=(0123)^{5{n-1}}=(0123). \end{aligned}$$

(17)

Hence, the permutation with one period is a 5-cycle:

$$\begin{aligned} \pi _{I_2^{[n]}}\pi _{I_1^{[n]}}=(0123)(04)=(04123), \end{aligned}$$

(18)

which guarantees the ergodicity in site (n+1). Furthermore, the structure (0, 1, 2, 3)(0, 4) is kept, because the state in island (I_1) is transferred to island (I_2) only when the state after (\pi _{I_1^{[n]}}) is 0 (see Fig. 9).

Fig. 9

The sequence of first, second, and third cells in Type09-2 with ((a,b)=(2,3)). The sequence in the third cell is divided into 5 lines that go from the top to the bottom. The light gray and dark gray regions represent islands (I_1) and (I_2), respectively. Two solid rectangles represent the